recall and use a = rω2 and a = v2 / r

Centripetal Acceleration

Learning Objective

Recall and use the two equivalent forms of the centripetal acceleration formula:

• $a = r\omega^{2}$
• $a = \dfrac{v^{2}}{r}$

Definition

When an object moves in a circular path of radius $r$ with constant speed, its velocity vector continuously changes direction. This change requires an acceleration directed towards the centre of the circle, called centripetal acceleration.

Derivation of the Formulas

1. Start from the definition of angular velocity: $\omega = \dfrac{v}{r}$.
2. Substituting $v = r\omega$ into the linear‑motion expression $a = \dfrac{v^{2}}{r}$ gives: $$a = \dfrac{(r\omega)^{2}}{r} = r\omega^{2}.$$
3. Conversely, solving $\omega = \dfrac{v}{r}$ for $v$ and substituting into $a = r\omega^{2}$ yields: $$a = r\left(\dfrac{v}{r}\right)^{2} = \dfrac{v^{2}}{r}.$$

Relationship Between Linear and Angular Quantities

The following table summarises the key relationships used in circular motion.

Quantity	Symbol	Formula
Radius of path	$r$	–
Linear speed	$v$	$v = r\omega$
Angular speed	$\omega$	$\omega = \dfrac{v}{r}$
Centripetal acceleration	$a$	$a = r\omega^{2} = \dfrac{v^{2}}{r}$

Worked Example

Problem: A car travels around a circular track of radius $50\ \text{m}$ at a constant speed of $20\ \text{m s}^{-1}$. Find the magnitude of its centripetal acceleration.

1. Identify the given values: $r = 50\ \text{m}$, $v = 20\ \text{m s}^{-1}$.
2. Use the formula $a = \dfrac{v^{2}}{r}$: $$a = \frac{(20\ \text{m s}^{-1})^{2}}{50\ \text{m}} = \frac{400\ \text{m}^{2}\text{s}^{-2}}{50\ \text{m}} = 8\ \text{m s}^{-2}.$$
3. Result: The car experiences a centripetal acceleration of $8\ \text{m s}^{-2}$ directed towards the centre of the track.

Common Mistakes to Avoid

• Confusing centripetal (towards the centre) with centrifugal (apparent outward) forces.
• Using the radius of curvature incorrectly; always use the radius of the actual circular path.
• Mixing units: ensure $v$ is in $\text{m s}^{-1}$, $r$ in metres, and $\omega$ in $\text{rad s}^{-1}$.
• For problems that give $\omega$, remember to convert to linear speed if required: $v = r\omega$.

Practice Questions

1. A stone is tied to a string and whirled in a horizontal circle of radius $0.75\ \text{m}$ at an angular speed of $12\ \text{rad s}^{-1}$. Calculate the centripetal acceleration of the stone.
2. A satellite orbits Earth in a circular orbit of radius $7.0\times10^{6}\ \text{m}$ with a speed of $7.5\times10^{3}\ \text{m s}^{-1}$. Determine the required centripetal acceleration.
3. Show that the period $T$ of uniform circular motion can be expressed as $T = 2\pi\sqrt{\dfrac{r}{a}}$ using $a = r\omega^{2}$.